Problem: What is the average rate of change of the function $g(x)=\cos(x)$ over the interval $2\leq x \leq 2+h$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{\cos(h)-\cos(2)}{h}$ (Choice B) B $\dfrac{\cos(2+h)-\cos(2)}{h}$ (Choice C) C $\dfrac{\cos(2+h)-\cos(2)}{2}$ (Choice D) D $\dfrac{\cos(2+h)-\cos(2)}{2+h}$
Solution: This is the formula for the average rate of change of a function $f$ over the interval $[a,b]$ : $\dfrac{f(b)-f(a)}{b-a}$ We are interested in the average rate of change of $g(x)=\cos(x)$ over the interval $2\leq x \leq 2+h$ : $\begin{aligned} &\phantom{=}\dfrac{g(2+h)-g(2)}{(2+h)-(2)} \\\\ &=\dfrac{\cos(2+h)-\cos(2)}{2+h-2} \\\\ &=\dfrac{\cos(2+h)-\cos(2)}{h} \end{aligned}$ The average rate of change of the function is $\dfrac{\cos(2+h)-\cos(2)}{h}$. Notice that the average rate of change is calculated just like the slope of the secant line that intersects the graph of the function at the interval's endpoints.